Question: The mayor of Brookmarsh is running a campaign to revitalize his city. Currently, the population of Brookmarsh is $10{,}000$ and is increasing at a rate of $2\%$ per year. The mayor predicts that the population will continue to grow in this manner, and that in $t$ years, the population will be at least $15{,}000$. Write an inequality in terms of $t$ that models the situation.
The strategy This problem involves a population increasing by $2\%$ every year. So, to find the population over time, we repeatedly multiply the initial population, $10{,}000$, by $1.02$. [Why?] Because of this, we know we can model the situation with an exponential expression of the form $ab^x$, where $a$ is $10{,}000$ and $b$ is $1.02$. We now only need to find $x$, which represents the number of times the population has increased by $2\%$. Finding the exponent We know that every year the population increases by $2\%$. Since time is in years, the exponent is simply $t$. Writing an inequality We can now replace $x$ in the original model with $t$. Therefore, the expression $10{,}000\cdot (1.02)^t$ models the population of Brookmarsh in $t$ years. Since the mayor predicts that the population in $t$ years is at least $15{,}000$, we can write the following inequality. $10{,}000\cdot (1.02)^t\geq15{,}000$ The answer An inequality that models the situation is $10000\cdot (1.02)^t\geq15000$.